Some analytical results for spin 1/2 spin systems. III: The benzene ring

In two earlier papers on multiply connectedABCD spin 1/2 spin systems, it was shown that it is possible to simplify the calculation of (i) the time dependent density matrix(t), and (ii) the time evolution of high-order multiple quantum operators, evolving in the presence of differing Zeeman offsets, scalar coupling and dipolar interactions, by subdividing the Hamiltonian into , where , is a suitable linear combination of the constants of the motion. In this paper, these techniques are applied to the benzene ring with particular emphasis on high-order multiple-quantum NMR experiments, and their interpretation in terms of specific wavefunctions. In particular, it is shown that excellent agreement between theory and experiment can be obtained for the energy splittings witnessed in the m = +4,+5 MQ-NMR transitions, with minimum effort.     

Some analytical results forABC,ABCD, andXBCD coupled spin 1/2 systems. I

By choosing a suitable linear combination of the constants of the motion , it is shown that the calculation of the density matrix(t) can be simplified by subdividing the Hamiltonian into . In particular, this technique can be used to obtain closed form solutions for the eigenfunctions and eigenvalues of spin 1/2ABC andXBCD spin systems, evolving in the presence of Zeeman offsets, scalar coupling and dipolar interactions. In general, the eigenvalues and eigenvalues of are very transparent, while those of require more effort. Nevertheless, simplifications can be made. Firstly, the effective size of the Hamiltonian matrix which needs to be considered, is reduced fromN ×N to at least (N – 2) × (N – 2), while forXBC ... systems it is reduced to (N – 4) × (N – 4). Secondly, the highest rank and highest/lowest order tensor operators available to the spin ensemble are constants of the motion under . Finally, by exploiting the fact that is a good quantum number, it is possible to block-diagonalize the matrix into no more than 3 × 3 matrices.     

Some analytical results forABC,ABCD, andXBCD coupled spin 1/2 systems. II

In the preceding paper, it was shown that the calculation of the density matrix(t) for multiply connectedABC, etc., spin 1/2 spin systems can be greatly simplified by subdividing the Hamiltonian into , where is a suitable linear combination of the constants of the motion. In this paper, a framework for the determination of the time evolution of high-order multipolar quantum states is presented and discussed. It is shown that the necessary mathematical labour is reduced to a minimum by (i) exploiting the fact that is a good quantum number, and (ii) using the theory of partitioned matrices. For example, it is shown that for a generaln-coupled spin 1/2 system, the spin dynamics of the multipolar states, whereK _max is the maximum tensorial rank, can be determined without the need to diagonalize the full 2 ⁿ × 2 ⁿ Hamiltonian matrix, wheren is the number of spins. In fact, to describe the time evolution of the multipolar states it is only necessary to diagonalize twon ×n matrices at most. Finally, some cautionary remarks are made concerning the use of the weak-coupling approximation.     

Irreducible half-integer rank unit spherical tensors

A new class of half-integer rank spherical tensors is introduced. The motivation for investigating this new class of tensors originated from a desire to be able to partition matrices using mixtures of fictitious integer and half-integer spin labels. However, it is shown that they can also be used as annihilation/creation operators for spin-1/2, 3/2, etc., particles. In particular, half-integer rank tensors can be used to add/subtract a spin-1/2 particle from a given ensemble. Thus they can be viewed as the natural generalization of the raising and lowering operatorsI _±, in that they change bothI andM, simultaneously.The concept of a universal rotator is introduced and it is demonstrated that half-integer rank tensors obey the same contractional and rotational properties as their integer counterparts, but with half-integer rank. In addition, it is shown that half-integer rank tensors can be used to factorize the Pauli spin matrices. Finally, an example of the use of half-integer rank tensors in the block-diagonalization of a simple 3 x 3 matrix is presented and discussed.     

\pi -Electron Long-Range Spin–Spin Coupling in the Full Configuration Interaction Method

A technique for calculating spin perturbation effects in the framework of the previously developed matrix version of the full configuration interaction method is presented. The method is adapted to calculations of -electron atom–atomic spin polarizability defined as the -contribution to the indirect spin–spin nuclear coupling constant. Particular calculations show that pronounced long-range spin–spin coupling is actually possible in some characteristic conjugated systems such as polymethine dye carbocations or some nonalternant hydrocarbons. At the same time, spin–spin correlators appearing in McKonnel's theory are inapplicable in describing such long-range effects.     

Principles of spin-echo modulation by J-couplings in magic-angle-spinning solid-state NMR.

In magic-angle-spinning solid-state NMR, the homonuclear J-couplings between pairs of spin-1/2 nuclei may be determined by studying the modulation of the spin echo induced by a pi-pulse, as a function of the echo duration. We present the theory of J-induced spin-echo modulation in magic-angle-spinning solids, and derive a set of modulation regimes which apply under different experimental conditions. In most cases, the dominant spin-echo modulation frequency is exactly equal to the J-coupling. Somewhat surprisingly, the chemical shift anisotropies and dipole-dipole couplings tend to stabilise--rather than abscure--the J-modulation. The theoretical conclusions are supported by numerical simulations and experimental results obtained for three representative samples containing 13C spin pairs.     

Relationship of magnetic resonance,thermodynamic, and structural characteristics of π complexes of free radicals with aromatic molecules

Methods of EPR spectroscopy and GLC were used to determine the magnetic resonance and thermodynamic characters of the formation of paramagnetic complexes consisting of a free radical (diphenyldicumynitroxyl, triphenylverdazyl, indophenoxyl) and substituted benzenes. It was established that in such systems, depending on the donor-acceptor and polar properties of the diamagnetic molecules, there form paramagnetic complexes of two types: donor-acceptor complexes in which the free radical acts as a -electron acceptor and both intramolecular and intermolecular redistribution of spin density is accomplished, and dipole-dipole complex in which electrostatic interaction determines mutual orientation of molecules in the complex and intramolecular distribution of electron and spin densities. Spectral and thermodynamic criteria of the formation of donor-acceptor and dipole-dipole paramagnetic free radical-diamagnetic molecule complexes are formulated.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 25, No. 3, pp. 283–289, May–June, 1989.     

Origin and meaning of the Fermi contact interaction

The Fermi-contact interaction (FCI) can easily be derived from 1st order perturbation theory applied to the non-relativistic wave equation for a spin-(1/2) particle of Lévy-Leblond, with the nuclear spin described by the field of an external magnetic dipole, and it results from the fact that the turn-over-rule for the operator is only valid if the derivatives implicit in are taken in the distribution sense. If one avoids to apply the turn-over-rule, the FCI is obtained without the need to introduce a -function. It is also shown that the formulation of a magnetic point dipole as the limit of an extended nucleus directly leads to the FCI. Traditional methods of the derivation of the FCI are analyzed in the light of this new interpretation. It is then explained why the perturbation expansions in powers of the magnetic moment of the nucleus necessarily diverges, but that the expression for the 1st order energy on which the concept of the FCI is based, can nevertheless be justified by means of the Hellmann-Feynman theorem with a correction term if singular wave functions are involved. Finally some comments on a theory beyond first order are made.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Spin–Spin Relaxation in the 17O NMR Spectra of Na+ and K+ Water Clusters

Relaxation characteristics of Na⁺ and K⁺ water clusters were studied by ¹⁷O NMR spectroscopy. The influence of viscosity and pH of solution was taken into account; for both electrolytes in the concentration range -4 M, spin–spin relaxation times are greater than those for pure water. For K⁺ clusters, maxima of the concentration dependence of ¹⁷O spin–spin relaxation time have been found.     

Handling multicomponent systems in \mathbb{R}^n. I: Theoretical results

The phases of multicomponent systems (mixtures, states, etc.) containing the compounds are , where and . For (quaternary or higher dimensional systems), the displaying methods and visual investigations in the dimensional Euclidean space are tangentially or not at all described in the literature. In this paper we first develop the theoretical (both mathematical and computational) background in any dimension in . We focus not only on the important points, lines, surfaces of these systems, and computing method of the states of some processes in such systems, but also on the approximating methods of the above mentioned lines and surfaces, and, finally, on the question which is the region where a state (a point) falls into. Using the above results a computer program for PC's was created for evaluating and displaying the approximated surfaces. This program is described in I. Szalkai, SALT3DIM.exe – A program for handling 4 component mixtures, Preprint No. 047, University of Veszprém (1996), and the computing results are planned to be published in a forthcoming paper (I. Szalkai, Handling multicomponent systems in . II: Computational results, J. Chem. Inf. Comput. Sci., submitted).     

Symmetry breaking in HF wave functions of Fe(CH)2

HF and CAS calculations for linear geometry of Fe(CH)₂ with symmetry have been performed. The basis sets used were DZ and DZ + P with ECP on the iron atom. Two closedshell and one quintet RHF wave functions have been found, and . All of them are singlet and triplet unstable in the wide range of Fe–CH distances. Singlet instability leads to the Charge Density Wave (CDW) brokensymmetry wave function with two electrons on carbon or orbital in the dissociation limit. Triplet instabilities lead to two brokensymmetry HF wave functions of Axial Spin Density Wave (ASDW) type, ASDW₁ and ASDW₂. In the dissociation limit they give carbon atoms with two electrons on and orbitals coupled to singlet and triplet, respectively. The stability conditions for CDW, ASDW₁ and ASDW₂ instabilities have been derived. Other HF wave functions with spin symmetry unrestricted have been also found. CAS(8,8), CAS(10,10) and CAS(12,12) calculations for singlet, triplet and quintet states of Fe(CH)₂ have been carried out. In all CAS calculations the singlet state has the lowest energy. The Fe–CH equilibrium distances obtained from closedshell RHF wave functions are much shorter and from brokensymmetry wave functions are much longer than those obtained from CAS calculations.     

Chemical shifts of F19 NMR in spectra of trifluorovinyl compounds

The regularities of the changes of the chemical shift in the F¹⁹ NMR of trifluorovinyl compounds are discussed in this paper. It has been shown that the conditions , where _F ⁰ is the shift of F¹⁹ in tetrafluorethylene (X=F) are satisfied for the chemical shifts of fluorine in compounds. An increase in the electronegativity of X leads to a decrease in the ionic nature of the C-F₃ bond and to a shift toward weak fields. The effect of X on the and shifts is not linked in a specific way to the electronegativity of the X group and is probably determined by the effect of the electric fields on the shielding of the F¹⁹ nucleus. Increments were calculated from the data obtained on the shifts in trifluorovinyl compounds. The method of increments was used for the assignment of the lines in 1, 2-difluoro olefins. This method for the assignment is confirmed by known data. Several cases for the application of the increment method are proposed. The results of spectral measurements of the F¹⁹ NMR in some trifluorovinyl compounds are given in the experimental part.     

Modeling of Thermodynamic Properties of Amino Acids and Peptides Using Additivity and HKF Theory

Relative densities, , and heat capacity ratios, of aqueous L-histidine, L-phenylalanine, L-tyrosine, L-tryptophan, and L-2,3-dihydroxyphenylalanine (L-dopa) have been measured at 15, 25, 40, and 55°C and 0.1 MPa. Apparent molar volumes, V _2,, apparent molar heat capacities, C_P2,, partial molar volumes at infinite dilution, , and partial molar heat capacities at infinite dilution, , have been calculated from these measurements and compared to available literature values. The partial molar properties at infinite dilution for these systems have been added to those previously obtained for amino acids and peptides in water and the combined set used as input to a novel additivity analysis. The model we develop is based upon the equations of state of Helgeson, Kirkham, and Flowers (HKF) and has been constructed with additive parameters. The model may be used to predict thermodynamic properties of many aqueous biochemicals over an extended temperature range. Group contributions to the parameters in our model and effective Born coefficients are reported for 24 aqueous amino acid and peptide systems. Our results are compared to data previously published in the literature.     

Volumetric Properties of Binary Mixtures of Acetonitrile and Chloroalkanes at 25°C and Atmospheric Pressure

Excess molar volumes for binary mixtures of acetonitrile + dichloromethane, acetonitrile + trichloromethane, and acetonitrile + tetracloromethane at 25°C have been used to calculate partial molar volumes , excess partial molar volumes , and apparent molar volumes of each component as a function of composition. The V _m ^Evalues are negative over the entire composition range for the systems studied. The applicability of the Prigogine–Flory–Patterson theory was explored. The agreement between theoretical and experimental results is satisfactory for the systems with dichloromethane and tetrachloromethane. For the unsymmetrical behavior of the system with trichloromethane, however, the agreement is poor.     

Preferential solvation in three component systems: Evaluation of Kirkwood-Buff parameters

The extent of local excess or deficiency of a component solvent near the solute in a mixed binary solvent has been calculated using the Hall formalism for the Kirkwood-Buff equation. The possibility of calculation of the two solute-solvent Kirkwood-Buff parameters using the values is discussed. A model calculation using literature data for preferential solvation in mixed binary solvents is presented. The solute-solvent and solvent-solvent interactions and the relative size of the solvents are also shown to be relevant factors in determining the values.     

A Thermodynamic Analysis of Ion Adsorption in the Metal Oxide/Electrolyte Systems in which PZC and CIP do not Coincide

A theoretical-numerical analysis of two adsorption systems composed of the same kind of oxide- TiO₂, and of two different electrolytes, NaCl and CsCl is presented. For one kind of the electrolyte (NaCl), PZC and CIP coincide, whereas they are different for the other (CsCl) electrolyte. The analysis is carried out by applying the popular TLM model, and by drawing formal-mathematical consequences of CIP existence in both kinds of adsorption systems. The values of the adsorption parameters are found by fitting simultaneously the obtained theoretical expressions to both experimental titration isotherms, and to the individual isotherms of cation adsorption measured using radiometric methods. That theoretical-numerical analysis suggests, that the inequality PZC (pK ^int _a1 + pK ^int _a2) may be a general feature of the oxide/electrolyte systems including the systems in which PZC and CIP coincide.     

Effect of solvent on the partial molal volumes and heat capacities of nonelectrolytes

Group contributions to in seven solvents and to in three solvents have been tabulated. The variation of group parameters is discussed in terms of the solvent compressibility coefficient, _T. The scaled particle theory (SPT) is used to calculate cavity contributions to and C _p2 ^o . Interaction contributions are obtained from the cavity terms and and values estimated through the additivity schemes. values are more sensitive to solute-solvent interactions than in water and less sensitive in methanol. The SPT results for heat capacities support the concept of structural promotion by hydrophobic solutes in water.     

The sorption of uranyl ions on silica-titania mixed-hydroxide gels

SiO₂–TiO₂ mixed hydroxide gels were prepared and the apparent adsorption capacity for Na⁺ (1.30–1.60 mmol/g dry sample) determined by pH titration. The effect of solution pH on the sorption ability of the binary gels was studied. It is found that: (1) the sorption mechanism depends upon the pH, and the two most important factors are the nature of both counter ions in bulk solution and the surface of silica-titania gels, (2) the best pH range for the sorption of uranyl ions is 4–7. The thermodynamic functions were evaluated, suggesting that the reaction is chiefly attributed to physical adsorption rather than chemical one, and the opposite is the case of the reaction.